Alan had some cards and he gave some to two friends, Ben and Carl.
Alan first gave 1/3 of his cards and 8 more cards to Ben.
Alan then gave 3/4 of the remainder to Carl and 2 more cards.
In the end, Alan was left with 46 cards. How many cards did Alan have at first?
Thanks for your help.
ben got 1/3 x + 8, leaving 2/3 x - 8
carl got 3/4 (2/3 x - 8) + 2
so, doing the math,
x - (1/3 x + 8) - 3/4 (2/3 x - 8) = 46
Now solve for x
200
Let's break down the information given step-by-step to find the starting number of cards Alan had:
1. Alan gave 1/3 of his cards and 8 more cards to Ben.
Let's assume the number of cards Alan had originally was "x."
Alan gave 1/3 * x cards to Ben.
He also gave an additional 8 cards to Ben.
So, Ben received (1/3 * x) + 8 cards.
After giving cards to Ben, Alan is left with x - ((1/3 * x) + 8) cards.
2. Alan gave 3/4 of the remainder to Carl and 2 more cards.
Alan gave 3/4 * (x - ((1/3 * x) + 8)) cards to Carl.
He also gave an additional 2 cards to Carl.
So, Carl received 3/4 * (x - ((1/3 * x) + 8)) + 2 cards.
After giving cards to Carl, Alan is left with [x - ((1/3 * x) + 8)] - [3/4 * (x - ((1/3 * x) + 8)) + 2] cards.
3. Finally, Alan was left with 46 cards.
We can now set up an equation to solve for the starting number of cards, x.
[x - ((1/3 * x) + 8)] - [3/4 * (x - ((1/3 * x) + 8)) + 2] = 46
Let's simplify the equation step-by-step:
[x - (1/3 * x + 8)] - [3/4 * (x - 1/3 * x - 8) + 2] = 46
[x - 1/3 * x - 8] - [3/4 * (2/3 * x - 8) + 2] = 46
[x - 1/3 * x - 8] - [3/4 * 2/3 * x - 3/4 * 8 + 2] = 46
[x - 1/3 * x - 8] - [1/2 * x - 3/2 + 2] = 46
Combining like terms:
[x - 1/3 * x - 8] - [1/2 * x + 1/2] = 46
(6/6 * x - 2/6 * x - 48/6) - (3/6 * x + 3/6) = 46
(4/6 * x - 48/6) - (3/6 * x + 3/6) = 46
(1/6 * x - 45/6) = 46
(1/6 * x) = 46 + 45/6
(1/6 * x) = 276/6 + 45/6
(1/6 * x) = 321/6
Multiplying both sides by 6:
x = 321
Therefore, Alan initially had 321 cards.
To solve this problem, let's break it down step by step:
Step 1: Alan first gave 1/3 of his cards and 8 more cards to Ben. Let's represent the number of cards Alan had originally as x.
After giving away 1/3 of his cards, Alan was left with 2/3 of x cards.
So, he gave away (1/3)x + 8 cards to Ben. This means Alan was left with (2/3)x - ((1/3)x + 8) cards.
Step 2: Alan then gave 3/4 of the remainder to Carl and 2 more cards. Let's denote the number of cards Alan had at this point as y.
After giving away 3/4 of the remaining cards, Alan was left with 1/4 of y cards.
So, he gave away (3/4)((2/3)x - ((1/3)x + 8)) + 2 cards to Carl. This means Alan was left with 1/4y - (3/4)((2/3)x - ((1/3)x + 8)) - 2 cards.
Step 3: In the end, Alan was left with 46 cards. According to Step 2, this means 1/4y - (3/4)((2/3)x - ((1/3)x + 8)) - 2 = 46.
Now, let's solve this equation to find the values of x and y, which represent the number of cards Alan had originally and at the second stage.
Step 4: Simplify the equation.
1/4y - (3/4)((2/3)x - ((1/3)x + 8)) - 2 = 46.
1/4y - (1/2)x + 4 - 2 = 46.
1/4y - (1/2)x + 2 = 46.
1/4y - (1/2)x = 44.
Step 5: Multiply through by 4 to clear fractions.
y - 2x = 176.
Step 6: Use the information from the problem that Alan was left with 46 cards in the end.
y = 46.
Step 7: Substitute y = 46 into the equation from Step 5.
46 - 2x = 176.
Step 8: Solve for x.
-2x = 130.
x = 130 / -2.
x = -65.
Hence, Alan had 65 cards at first.